### Paper 2

**Instructions to candidates**

- Write your session number in the boxes above.
- Do not open this examination paper until instructed to do so.
- A graphic display calculator is required for this paper.
- Section A: answer all questions. Answers must be written within the answer boxes provided.
- Section B: answer all questions in the answer booklet provided. Fill in your session number on the front of the answer booklet, and attach it to this examination paper and your cover sheet using the tag provided.
- Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures.
- A clean copy of the mathematics HL and further mathematics HL formula booklet is required for this paper.
- The maximum mark for this examination paper is [100 marks].
- Time: 120 minutes

**Topic 1: Number and algebra****– **SL content

- Topic : SL 1.1
- Topic : SL 1.2
- Topic : SL 1.3Â
- Topic : SL 1.4
- Topic : SL 1.5
- Laws of exponents with integer exponents.
- Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.

- Topic : SL 1.6
- Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity.

- Topic : SL 1.7
- Laws of exponents with rational exponents.
- Laws of logarithms.
- log
_{a}xy = log_{a}x + log_{a}y - loga\(\frac{x}{y}\)=logax – log
_{a}y - log
_{a}x^{m}= mlog_{a}x for a, x, y > 0

- log
- Change of base of a logarithm.
- log
_{a}x = \(\frac{log_bx}{log_ba}\)Â for a, b, x > 0

- log
- Solving exponential equations, including using logarithms

- Topic : SL 1.8
- Sum of infinite convergent geometric sequences.

- Topic : SL 1.9

**Topic 1: Number and algebra****– AH**L content

- Topic : AHL 1.10
- Topic : AHL 1.11
- Partial fractions

- Topic : AHL 1.12
- Topic : AHL 1.13
- Topic : AHL 1.14
- Topic : AHL 1.15
- Topic : AHL 1.16

### Topic 2: Functions**– **SL content

- Topic: SL 2.1
- Different forms of the equation of a straight line.
- Gradient; intercepts.
- Lines with gradients m
_{1}and m_{2} - Parallel lines m
_{1}= m_{2}. - Perpendicular lines m
_{1}Ã— m_{2}= âˆ’ 1.

- Topic: SL 2.2
- Concept of a function, domain, range and graph. Function notation, for example f(x), v(t), C(n). The concept of a function as a mathematical model.
- Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line y = x, and the notation f
^{âˆ’1}(x).

- Topic: SL 2.3
- The graph of a function; its equation \(y = f\left( x \right)\) .
- Creating a sketch from information given or a context, including transferring a graph from screen to paper. Using technology to graph functions including their sums and differences.

- Topic: SL 2.4
- Topic: SL 2.5
- Composite functions.
- (f âˆ˜ g)(x) = f(g(x))

- Identity function.
- Finding the inverse function f
^{âˆ’1}(x)- (f âˆ˜ f
^{âˆ’1})(x) = (f^{âˆ’1}âˆ˜ f)(x) = x

- (f âˆ˜ f

- Composite functions.
- Topic: SL 2.6
- The quadratic function f(x) = ax
^{2}+ bx + c: its graph, y -intercept (0, c). Axis of symmetry. - The form f(x) = a(x âˆ’ p)(x âˆ’ q), x intercepts (p, 0) and (q, 0).The form f(x) = a (x âˆ’ h) 2 + k, vertex (h,k).

- The quadratic function f(x) = ax
- Topic: SL 2.7
- Topic: SL 2.8
- The reciprocal function f(x) = 1 x , x â‰ 0: its graph and self-inverse nature.
- The rational function \(f(x)=\frac{{ax + b}}{{cx + d}}\)and its graph. Equations of vertical and horizontal asymptotes.

- Topic: SL 2.9
- Exponential functions and their graphs
- The function \(f(x)=a^x , a>0,f(x)=e^x\), \(a > 0\) , and its graph.

- Logarithmic functions and their graphs:
- The function \(f(x)=log_ax,x>0,f(x)=lnx,x>0\) , and its graph

- Exponential functions and their graphs
- Topic: SL 2.10
- Solving equations, both graphically and analytically.
- Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
- Applications of graphing skills and solving equations that relate to real-life situations

- Topic SL 2.11
- Transformations of graphs.
- Translations: y = f(x) + b; y = f(x âˆ’ a).
- Reflections (in both axes): y = âˆ’ f(x); y = f( âˆ’ x).
- Vertical stretch with scale factor p: y= p f(x).
- Horizontal stretch with scale factor \(\frac{1}{q}\): y = f(qx).

- Composite transformations.

- Transformations of graphs.

### Topic 2: Functions**– **AHL content

- Topic: AHL 2.12
- Topic: AHL 2.13
- Rational functions of the form
- \(f(x)=(\frac{ax + b}{cx^2 + dx + e}),and \; f(x)=\frac{ax^2 + bx + c}{dx + e}\)

- Rational functions of the form
- Topic: AHL 2.14
- Topic: AHL 2.15
- Topic: AHL 2.16

### Topic 3: Geometry and trigonometry-SL content

- Topic : SL 3.1
- The distance between two points in three dimensional space, and their midpoint.
- Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.
- The size of an angle between two intersecting lines or between a line and a plane.

- Topic SL 3.2Â
- Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
- The sine rule including the ambiguous case.
- \(\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}\)

- The cosine rule.
- \(c^2 = a^2 +b^2-2abcosC;\)
- \(cosC =\frac{a^2+ b^2-c^2}{2ab}\)

- Area of a triangle as \(\frac{1}{2}ab\sin C\) .

- Topic SL 3.3
- Applications of right and non-right angled trigonometry, including Pythagorasâ€™s theorem.
- Angles of elevation and depression.
- Construction of labelled diagrams from written statements.

- Topic SL 3.4
- Topic SL 3.5
- Definition of \(\cos \theta \) , \(\sin \theta \) in terms of the unit circle and \(\tan \theta \)Â as \(\frac{sin\theta }{cos\theta }\).
- Exact values of \(\sin\), \(\cos\) and \(\tan\) of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) and their multiples.
- Extension of the sine rule to the ambiguous case

- Topic SL 3.6
- Pythagorean identities: \({\cos ^2}\theta + {\sin ^2}\theta = 1\) ;
- Double angle identities for sine and cosine
- The relationship between trigonometric ratios.

- Topic : SL 3.7
- The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs
- Composite functions of the formÂ f(x) = asin(b(x + c)) + d.
- Transformations.
- Real-life contexts.
- height of tide, motion of a Ferris wheel.

- Topic : SL 3.8

### Topic 3: Geometry and trigonometry-AHL content

- Topic : AHL 3.9
- Topic : AHL 3.10
- Topic : AHL 3.11
- Relationships between trigonometric functions and the symmetry properties of their graphs.

- Topic : AHL 3.12
- Concept of a vector; position vectors; displacement vectors.
- Representation of vectors using directed line segments.
- Base vectors i, j, k.
- Components of a vector: \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .
- Algebraic and geometric approaches to the following:
- sum and difference of two vectors.
- the zero vector \(0\), the vector \( – v\) .
- multiplication by a scalar, \(kv\) , parallel vectors
- magnitude of a vector, \(\left| v \right|\) .unit vectors=\(\frac{\vec{v}}{\left | \vec{v} \right |}\)
- position vectors \(\overrightarrow {OA} = a\) .
- displacement vector \(\overrightarrow {AB} = b – a\) .

- Proofs of geometrical properties using vectors.

- Topic : AHL 3.13
- The definition of the scalar product of two vectors.
- Properties of the scalar product: \({\boldsymbol{v}} \cdot {\boldsymbol{w}} = {\boldsymbol{w}} \cdot {\boldsymbol{v}}\) ; \({\boldsymbol{u}} \cdot \left( {{\mathbf{v}} + {\boldsymbol{w}}} \right) = {\boldsymbol{u}} \cdot {\boldsymbol{v}} + {\boldsymbol{u}} \cdot {\boldsymbol{w}}\) ; \(\left( {k{\boldsymbol{v}}} \right) \cdot {\boldsymbol{w}} = k\left( {{\boldsymbol{v}} \cdot {\boldsymbol{w}}} \right)\) ; \({\boldsymbol{v}} \cdot {\boldsymbol{v}} = {\left| {\boldsymbol{v}} \right|^2}\) .
- The angle between two vectors.
- Perpendicular vectors; parallel vectors.

- Topic : AHL 3.14
- Topic : AHL 3.15
- Topic : AHL 3.16
- The definition of the vector product of two vectors.
- Properties of the vector product: \({\text{v}} \times {\text{w}} = – {\text{w}} \times {\text{v}}\) ; \({\text{u}} \times ({\text{v}} + {\text{w}}) = {\text{u}} \times {\text{v}} + {\text{u}} \times {\text{w}}\) ; \((k{\text{v}}) \times {\text{w}} = k({\text{v}} + {\text{w}})\) ; \({\text{v}} \times {\text{v}} = 0\) .
- Geometric interpretation of \({\text{v}} \times {\text{w}}\) .

- Topic : AHL 3.17
- Topic : AHL 3.18

**Topic 4 : Statistics and probability-SL content**

- Topic: SLÂ 4.1
- Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
- Reliability of data sources and bias in sampling.
- Interpretation of outliers.
- Sampling techniques and their effectiveness

- Topic: SLÂ 4.2
- Presentation of data (discrete and continuous): frequency distributions (tables).
- Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).
- Production and understanding of box and whisker diagrams.

- Topic: SLÂ 4.3
- Measures of central tendency (mean, median and mode).
- Estimation of mean from grouped data.
- Modal class.
- Measures of dispersion (interquartile range, standard deviation and variance).
- Effect of constant changes on the original data.
- Quartiles of discrete data.

- Topic: SLÂ 4.4
- Linear correlation of bivariate data. Pearsonâ€™s product-moment correlation coefficient, r.
- Scatter diagrams; lines of best fit, by eye, passing through the mean point.
- Equation of the regression line of y on x.
- Use of the equation of the regression line for prediction purposes.
- Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.

- Topic: SLÂ 4.5
- Topic: SLÂ 4.6
- Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
- Combined events: P(A âˆª B) = P(A) + P(B) âˆ’ P(A âˆ© B).
- Mutually exclusive events: P(A âˆ© B) = 0.
- Conditional probability; the definition \(P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\).
- Independent events; the definition \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)\) .

- Topic: SLÂ 4.7
- Topic: SLÂ 4.8
- Topic: SLÂ 4.9
- Topic: SLÂ 4.10
- Equation of the regression line of x on y.
- Use of the equation for prediction purposes.

- Topic: SLÂ 4.11
- Formal definition and use of the formulae:
- \(P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\).
- \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)\) .

- Formal definition and use of the formulae:
- Topic: SLÂ 4.12
- Standardization of normal variables (z- values).
- Inverse normal calculations where mean and standard deviation are unknown.

**Topic 4 : Statistics and probability-AHL content**

- Topic: AHLÂ 4.13
- Topic: AHLÂ 4.14

### Topic 5: Calculus-SL content

- Topic SL 5.1
- Topic SL 5.2
- Increasing and decreasing functions.
- Graphical interpretation of f â€²(x) > 0, f â€²(x) = 0, f â€²(x) < 0.

- Topic SL 5.3
- Derivative of f(x) = ax
^{n}is f â€²(x) = anx^{nâˆ’1}, n âˆˆ â„¤ - The derivative of functions of the form f(x) = ax
^{n}+ bx^{nâˆ’1}. . . . where all exponents are integers.

- Derivative of f(x) = ax
- Topic SL 5.4
- Topic: SL 5.5
- Introduction to integration as anti-differentiation of functions of the form f(x) = ax
^{n}+ bx^{nâˆ’1}+ …., where n âˆˆ â„¤, n â‰ âˆ’ 1. - Anti-differentiation with a boundary condition to determine the constant term.
- Definite integrals using technology.
- Area of a region enclosed by a curve y = f(x) and the x -axis, where f(x) > 0.

- Introduction to integration as anti-differentiation of functions of the form f(x) = ax
- Topic: SL 5.6
- Topic: SL 5.7
- Topic: SL 5.8
- Topic SL 5.9
- Topic SL 5.10
- Topic SL 5.11

### Topic 5: Calculus-AHL content

- Topic: AHL 5.12
- Informal understanding of continuity and differentiability of a function at a point.
- Understanding of limits (convergence and divergence).
- Definition of derivative from first principles as \(f’\left( x \right) = \mathop {\lim }\limits_{h \to 0} {\frac{{f\left( {x + h} \right) – f\left( x \right)}}{h}} \).

- Higher derivatives.

- Topic: AHL 5.13
- The evaluation of limits of the form \(\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}}\) and \(\mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{{g\left( x \right)}}\) .using lâ€™HÃ´pitalâ€™s rule or the Maclaurin series.
- Repeated use of lâ€™HÃ´pitalâ€™s rule

- Topic: AHL 5.14
- Topic: AHL 5.15
- Derivatives of \(\tan x\), \(\sec x\) , cosec x , \(\cot x\) , \({a^x}\) , \({\log _a}x\) , \(\arcsin x\) , \(\arccos x\) and \(\arctan x\) .
- Indefinite integrals of the derivatives of any of the above functions. The composites of any of these with a linear function.Â
- Use of partial fractions to rearrange the integrand.

- Topic: AHL 5.16
- Integration by substitution.
- Integration by parts.
- Repeated integration by parts.

- Topic: AHL 5.17
- Topic: AHL 5.18
- Topic: AHL 5.19